Super fiber bundles, connection forms, and parallel transport
- 1 June 2021
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (6), 063506
- https://doi.org/10.1063/5.0044343
Abstract
The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.Funding Information
- Studienstiftung des Deutschen Volkes
This publication has 39 references indexed in Scilit:
- MacDowell–Mansouri gravity and Cartan geometryClassical and Quantum Gravity, 2010
- Infinite-Dimensional and Colored SupermanifoldsJournal of Non-linear Mathematical Physics, 2010
- Some properties of the Noether charge and a proposal for dynamical black hole entropyPhysical Review D, 1994
- A global theory of supermanifoldsJournal of Mathematical Physics, 1980
- INTRODUCTION TO THE THEORY OF SUPERMANIFOLDSRussian Mathematical Surveys, 1980
- Two approaches to supermanifoldsTransactions of the American Mathematical Society, 1980
- The structure of supermanifoldsTransactions of the American Mathematical Society, 1979
- Unified Geometric Theory of Gravity and SupergravityPhysical Review Letters, 1977
- Unified Geometric Theory of Gravity and SupergravityPhysical Review Letters, 1977
- Progress toward a theory of supergravityPhysical Review D, 1976